Abstract

At academic teaching hospitals around the country, the majority of clinical care is provided by resident physicians. During their training, medical residents often rotate through various hospitals and/or medical services to maximize their education. Depending on the size of the training program, manually constructing such a rotation schedule can be cumbersome and time consuming. Further, rules governing allowable duty hours for residents have grown more restrictive in recent years (ACGME 2011), making day-to-day shift scheduling of residents more difficult (Connors et al., J Thorac Cardiovasc Surg 137:710–713, 2009; McCoy et al., May Clin Proc 86(3):192, 2011; Willis et al., J Surg Edu 66(4):216–221, 2009). These rules limit lengths of duty periods, allowable duty hours in a week, and rest periods, to name a few. In this paper, we present two integer programming models (IPs) with the goals of (1) creating feasible assignments of residents to rotations over a one-year period, and (2) constructing night and weekend call-shift schedules for the individual rotations. These models capture various duty-hour rules and constraints, provide the ability to test multiple what-if scenarios, and largely automate the process of schedule generation, solving these scheduling problems more effectively and efficiently compared to manual methods. Applying our models on data from a surgical residency program, we highlight the infeasibilities created by increased duty-hour restrictions placed on residents in conjunction with current scheduling paradigms.

Keywords

Notes

Acknowledgements

We would like to express our thanks to Dr. Keith Delman and Dr. Jeffrey Jopling for their assistance and support of this research. We are also grateful to Dr. Ozlem Ergun, Busra Ergun, Lesley-Anne Harris, and Kevin Yee for their contributions in the earlier stages of this study.

This work was supported by the William B. George Health Systems Institute Fellowship, the Mary Anne and Harold R. Nash Endowment, the Joseph C. Mello Endowment, and the Smalley Endowment at Georgia Tech.

Appendix

Sets, parameters, and decision variables used by RRA-IP are defined in Tables 7, 8 and 9, respectively.

The following example is provided to give clarity to a few selected constraints. Recall that constraints (5), (6), and (7) in RRA-IP are the equivalence constraints. Consider the following simple example: A set of 6 residents in R1, 3 periods in P1, one resident group in each of \(\bar {I}_{1}\), \(\bar {I}_{2}\), and \(\bar {I}_{3}\) such that \(\bar {I}_{1}=\left \{1\right \}\), \(\bar {I}_{2}=\left \{2\right \}\), \(\bar {I}_{3}=\left \{3\right \}\), with residents 1 and 2 in G1, residents 3 and 4 in G2, and residents 5 and 6 in G3. Therefore, residents 1 and 2 have identical service requirements, residents 3 and 4 have identical cluster requirements, and residents 5 and 6 require similar cluster assignments. Let there be only two services 1 and 2, both contained in the same cluster c (so SSc = {1,2}). Then constraints (5), (6), and (7) become:

Constraints (11) and (12) in RRA-IP represent whether multiple assignments to the same service can be consecutive or not. Increasing the number of time periods in the above example to 4 and given the additional information that if a resident is assigned multiple times to the same service, then service 1 requires consecutive assignments and service 2 requires nonconsecutive assignments, then for resident 1 in R1, constraints (11) and (12) become:

Finally, constraints (13) limit multiple assignments to the same service within a given time period. If we increase the number of time periods in P1 to 6, and supposing that service 3 requires assignments at least 4 months apart, then for resident 1 in R1, constraints (13) become:

A.2 Discussion of differences in test solutions

The solution achieved with Model-B (i.e., balancing constraints and corresponding objective) had 1 fewer demand violation than the solution from Model-E (i.e., equivalence constraints), and 4 fewer request violations. PGY1-Prelim residents had fewer total assignments, but the number of assignments per resident were equal as opposed to the solution with Model-E. PGY1-Urol residents and PGY2-Prelim residents had more total assignments, and assignments per resident were equal for each group with either Model-B or Model-E. PGY2-Categorical residents had fewer total assignments, and assignments per resident were not equal as opposed to the solution with Model-E. The remaining resident groups had equal total numbers of assignments (and equal numbers of assignments per resident) with Model-E and Model-B. There were some differences in assignments to service clusters for each resident group, other than PGY3, PGY4, and PGY5, and the solution produced with Model-E included 5 instances of assignments to service clusters by residents in a resident group where equivalence was not achieved across all residents within that group, as compared to only 2 instances in the solution produced with Model-B.

Test 2

All resident groups require similar experience

The solution achieved with Model-B had an identical number of demand violations to the solution from Model-E, and 2 fewer request violations. PGY1-Categorical residents had fewer total assignments, but the number of assignments per resident were equal with Model-B and Model-E. PGY1-Prelim residents and PGY1-Urol residents had more total assignments, and the number of assignments per resident (by group) were equal as opposed to the solution with Model-E. PGY2-Categorical residents had fewer total assignments, and assignments per resident were not equal as opposed to the solution with Model-E. The remaining resident groups had equal total number of assignments (and equal numbers of assignments per resident) with Model-B and Model-E. There were some differences in assignments to service clusters for each resident group, other than PGY3 and PGY5, and both solutions included 8 instances of assignments to service clusters by residents in a resident group where equivalence was not achieved across all residents within that group.

Test 3

All resident groups require identical services

The solution achieved with Model-B had 7 more demand violations than the solution from Model-E, and an identical number of request violations. PGY1-Prelim residents had fewer total assignments, PGY1-Urol residents had more total assignments, but assignments per resident were equal for each group with either Model-B or Model-E. The remaining resident groups had equal total numbers of assignments (and equal numbers of assignments per resident) with Model-B and Model-E. There were some differences in assignments to service clusters for most resident groups, and both solutions included 0 instances of assignments to service clusters by residents in a resident group where equivalence was not achieved across all residents within that group.

Test 4

All resident groups require identical experience

The solution achieved with Model-B had 5 fewer demand violations than the solution from Model-E, and an identical number of request violations. PGY1-Categorical residents, PGY1-Urol residents, and PGY2-Prelim residents had more total assignments, PGY1-Prelim residents had fewer total assignments, but for each group, assignments per resident were equal with either Model-B or Model-E. The remaining resident groups had equal total numbers of assignments (and equal numbers of assignments per resident) with Model-B and Model-E. There were some differences in assignments to service clusters for some resident groups, and both solutions included 0 instances of assignments to service clusters by residents in a resident group where equivalence was not achieved across all residents within that group.

Test 5

No similar/identical experience/service requirements

The solution achieved with Model-B had an identical number of demand and request violations to the solution from Model-E, and demand violations were identical by service and time period. PGY2-Categorical residents had fewer total assignments, and assignments per resident were equal with either Model-B or Model-E. PGY2-Prelim residents had more total assignments, and assignments per resident were equal with Model-B unlike the solution found with Model-E. The remaining resident groups had equal total numbers of assignments (and equal numbers of assignments per resident) with Model-B or Model-E. The number of assignments to each service cluster was almost identical for each resident group, but both solutions included 59 instances of assignments to service clusters by residents in a resident group where equivalence was not achieved across all residents within that group.

Test 6

PGY1 require similar experience, PGY2-5 require identical experience

The solution achieved with Model-B had an identical number of demand and request violations to the solution from Model-E. PGY1-Prelim residents had fewer total assignments, and assignments per resident were equal with either Model-B or Model-E. PGY1-Urol residents had more total assignments, and assignments perresident were equal with Model-B unlike the solution found with Model-E. T he remaining resident groups had equal total numbers of assignments (and equal numbers of assignments per resident) with Model-B or Model-E. There were some differences in assignments to service clusters for PGY1 resident groups, and the solution produced with Model-E included 1 instance of assignments to service clusters by residents in a resident group where equivalence was not achieved across all residents within that group, as compared to 0 instances in the solution produced with Model-B.

Resident rotation assignment decision support tool - screenshot of service data entry form

Table 10

Services staffed by surgical residents from Emory University School of Medicine, with monthly demand requirements. GMH = Grady Memorial Hospital, EUH = Emory University Hospital, EMH = Emory University Hospital Midtown, VAH = Veterans Affairs Hospital, PGH = Piedmont General Hospital, HEH = Emory University Hospital at Egleston. An asterisk next to a resident level means: (i) if contained in an “or” statement, then the asterisk means that this level is preferred over the other, (ii) if not contained in an “or” statement, then the asterisk means that the level is preferred, but not demanded, (iii) if next to a number, then the asterisk means that that number of a particular resident level is preferred, but not demanded

Rotation

Service Name

Demand

1

GMH General Surgery A

1 PGY-1, 1 PGY-3* or PGY-4, and 1 PGY-5

1

GMH General Surgery B

1 PGY-1, 1 PGY-3* or PGY-4, and 1 PGY-5

1

EUH General Surgery A

1 PGY-1, 1 PGY-2 or PGY-3*, and 1 PGY-4 or PGY-5*

1

EUH General Surgery B

1 PGY-1 or PGY-2*, and 1 PGY-4 or PGY-5*

1

EUH ACS

1 PGY-1 or PGY-2*, and 1 PGY-4

1

EUH Surgery Oncology

1 PGY-1, 1 PGY-3, 1 PGY-4, and 1 PGY-5

1

EMH General Surgery

1 PGY-1, 1 PGY-2, and 1 PGY-4

1

PGH General Surgery

1 PGY-1*, 1 PGY-2*, 1 PGY-3*, 1 PGY-4*, and 1 PGY-5*

1

VAH General Surgery

1 PGY-2 and 1 PGY-4* or PGY-5

2

EUH Selective

1 PGY-3* and 1 PGY-5*

3

GMH Trauma Day

1 PGY-1

4

GMH Trauma Night

1 PGY-3, and 1 PGY-4

5

EUH SICU

1 PGY-2* or PGY-3

6

GMH SICU

1 PGY-1 and 2 or 3* PGY-2’s

7

EUH Vascular Surgery

1 PGY-1, 1 PGY-2, and 1 PGY-4* or PGY-5

8

VAH Vascular Surgery

1 PGY-2, and 1 PGY-3* or PGY-4

9

HEH Pediatric Surgery Day

3 PGY-1’s and 1 PGY-4

10

GMH Burns

1 PGY-1, 1 PGY-2, and 1 PGY-3

11

GMH Plastic Surgery

1 PGY-1

12

GMH Urology

1 PGY-1

13

EUH Transplant

1 PGY-1* or PGY-2 and 1 PGY-3 or PGY-4*

14

EUH Cardiothoracic

1 PGY-2

15

EUH Night

1 PGY-1 and 1 PGY-3

16

HEH Pediatric Surgery Night

1 PGY-1

17

VAH Cardiothoracic

1 PGY-2

18

GMH Quad Night

1 PGY-1

19

EUH Colorectal

1 PGY-1, 1 PGY-2, and 1 PGY-5

20

GMH General Surgery C

1 PGY2 and 1 PGY4

A.4 Surgical resident shift scheduling model (SRSS-IP) definitions

Sets, parameters, and decision variables used in SRSS-IP are presented in Tables 11, 12, and 13, respectively.

Table 11

Surgical Resident Shift Scheduling Model - Sets and descriptions

Set

Description

R

Set of all 1st-year medical residents assigned to the Emory group

S

Set of services in the group (i.e., S = {1 (EUH General Surgery A), 2 (EUH General Surgery B), 3 (EUH Surgery Oncology),